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arXiv:quant-ph/9611016 · Physical Review A

Dynamical reduction theory of Einstein-Podolsky-Rosen correlations and a possible origin of CP violations

Daniel I. Fivel

Published Jul 1, 1997• 15 citations
arXivHTMLPDF

Abstract

We show that there is essentially only one way to construct a stochastic Schr\"odinger equation that gives a dynamical account of the transformation of entangled into factorized states and is consistent with both quantum mechanics and required symmetries. The noisy, nonlinear term is a unimodular scalar multiple of the time-reversal operator that must be present whenever a Hamiltonian term in the Schr\"odinger equation can distinguish the factorized constituents of an entangled state. The dynamical mechanism involved in the transformation of entangled into factorized states provides an explanation for the fact that Einstein-Podolsky-Rosen correlations appear in a time determined by the response of the measuring device and independent of the distance between the particles. The dependence on the response time of the measuring device may be testable through a delay in observing the collapse of mesoscopic ``Schr\"odinger cat'' states in ion traps. It is further shown that there are situations where a two-particle interaction can induce a nonlinear term by virtue of coupling to decay modes that distinguish factorized constituents of an entangled state. We show that this should happen in the neutral $K$-meson system where the entangled ${K}_{L}$ state is pushed slightly in the direction of a factorized constituent ${(K}_{0}$ or $\overline{{K}_{0}})$ as a consequence of the fact that these can be distinguished via the sign of the charged lepton in a semileptonic decay mode. The result is a $\mathrm{CP}$ violation that is within 20% of the experimental value.

DOI
10.1103/physreva.56.146

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We show that there is essentially only one way to construct a stochastic Schr"odinger equation that gives a dynamical account of the transformation of entangled into factorized states and is consistent with both quantum mechanics and required symmetries. The noisy, nonlinear term is a unimodular scalar multiple of the time-reversal operator that must be present whenever a Hamiltonian term in the Schr"odinger equation can distinguish the factorized constituents of an entangled state. The dynamical mechanism involved in the transformation of entangled into factorized states provides an explanation for the fact that Einstein-Podolsky-Rosen correlations appear in a time determined by the response of the measuring device and independent of the distance between the particles. The dependence on the response time of the measuring device may be testable through a delay in observing the collapse of mesoscopic ``Schr"odinger cat'' states in ion traps. It is further shown that there are situations where a two-particle interaction can induce a nonlinear term by virtue of coupling to decay modes that distinguish factorized constituents of an entangled state. We show that this should happen in the neutral KKK-meson system where the entangled KL{K}_{L}KL​ state is pushed slightly in the direction of a factorized constituent (K0{(K}_{0}(K0​ or K0‾)\overline{{K}_{0}})K0​​) as a consequence of the fact that these can be distinguished via the sign of the charged lepton in a semileptonic decay mode. The result is a CP\mathrm{CP}CP violation that is within 20% of the experimental value.
We establish the precise connection between the theoretical T, CP, CPT asymmetries, in terms of transition probabilities between the neutral meson B d states, and the experimental asymmetries, in terms of the double decay rate intensities for Flavour-CP eigenstate decay products in a B-Factory of entangled states. Genuine asymmetry parameters in the time distribution of the asymmetries are identified and their measurability analysed, disentangling genuine and possible fake terms. The nine asymmetry parameters – three different observables for each one of the three symmetries – are expressed in terms of the ingredients of the Weisskopf-Wigner dynamical description of the entangled B d -meson states and we obtain a global fit to their values from the BaBar Collaboration experimental results. The possible fake terms are all compatible with zero. The information content of the nine asymmetry parameters is indeed different. The non-vanishing and are impressive separate direct evidence of Time-Reversal-Violation and CP-Violation in these transitions (and compatible with Standard Model expectations). A 2σ effect for the Re(θ) parameter responsible of CPT-Violation appears; interpreted as an upper limit, it leads to eV at 95% C.L. for the diagonal flavour terms of the mass matrix. Finally, we consider scenarios where, in the presence of quantum gravity fluctuations (space-time foam), the CPT operator may be ill-defined. Its perturbative treatment leads to a modification of the Einstein-Podolsky-Rosen correlation of the neutral meson system by adding an Entanglement-weakening term of the wrong exchange symmetry, the ω -effect. The analysis is extended to identify how to probe the complex ω when the connection between the Intensities for the two time-ordered decays ( f, g ) and ( g, f ) is lost ( f flavour and g CP eigenstate decay channels), and how the ω -effect is disentangled from CPT violation in the evolution Hamiltonian.
In a ϕϕϕ factory, neutral kaons are produced in Einstein-Podolsky-Rosen (EPR) states, furthermore, a KSK_SKS​ is tagged if its entangled partner is detected as KLK_LKL​, and vice versa. Here we point out that upon its detection as KLK_LKL​, one of the EPR-entangled particles is projected into one of the orthogonal basis states, consequently the other particle should be projected to the state orthogonal to KLK_LKL​, denoted as K⊥LK_{\perp L}K⊥L​, rather than KSK_SKS​, because KSK_SKS​ is not orthogonal to KLK_LKL​. Therefore the semileptonic charge asymmetry of what was previously regarded as KSK_SKS​ is really that of K⊥LK_{\perp L}K⊥L​. Based on this re-tagging, we modify the experimentally determined values of several parameters, including the one describing the violation of ΔS=ΔQΔS = ΔQΔS=ΔQ rule, the one describing CPT violation for ΔS=ΔQΔS = ΔQΔS=ΔQ transitions, and the time-like CPT- and Lorentz-violating coefficient for the two valence quarks in a kaon.
The quantum interference between the decays of entangled neutral kaons is a very powerful tool for testing the quantum coherence of the entangled kaon pair state. The studied process ϕ→KSKL→π+π−π+π−\phi\rightarrow K_SK_L\rightarrow\pi^+\pi^-\pi^+\pi^-ϕ→KS​KL​→π+π−π+π− exhibits the characteristic Einstein-Podolsky-Rosen correlations that prevent both kaons to decay into π+π−\pi^+\pi^-π+π− at the same time. The newly published result is based on data sample collected with the KLOE detector at DAΦ\PhiΦNE and corresponds to an integrated luminosity of about 1.7 fb−1^{-1}−1 , i.e. to ∼\sim∼ 1.7 ×\times× 109^99 ϕ→KSKL\phi\rightarrow K_SK_Lϕ→KS​KL​ decays. From the fit to the observed time difference distribution of the two kaon decays, the decoherence and CPT violation parameters of various phenomenological models are measured. A stringent upper limit on the branching ratio of the ϕ→KSKS,KLKL\phi \rightarrow K_S K_S , K_L K_Lϕ→KS​KS​,KL​KL​ decay is also derived. Independently, the comparison of neutral meson transition rates between flavour and CP eigenstates allows direct and model independent tests of time-reversal T and CPT symmetries to be conducted, through ratios of rates of two classes of processes: KSKL→(π±e∓ν)(3π0)K_S K_L \to (\pi^{\pm}e^{\mp}\nu)(3\pi^0)KS​KL​→(π±e∓ν)(3π0) and KSKL→(π+π−)(π±e∓ν)K_S K_L \to (\pi^{+}\pi^{-})(\pi^{\pm}e^{\mp}\nu)KS​KL​→(π+π−)(π±e∓ν). In addition to this a straightforward extension to the case of CPT symmetry is performed providing the first model independent test of CPT symmetry violation in transitions of neutral kaons.
The signal of ill-defined CPT weakening entanglement in the Bd{B_d}Bd​ B d system
In the presence of quantum-gravity fluctuations (space-time foam), the CPT operator may be ill-defined. Its perturbative treatment leads to a modification of the Einstein–Podolsky–Rosen correlation of the neutral meson system by adding an entanglement-weakening term of the wrong exchange symmetry, the ω\omega ω -effect. In the current paper we identify how to probe the complex ω\omega ω in the entangled BdB_dBd​ -system using the flavour (f)–CP(g) eigenstate decay channels: the connection between the intensities for the two time-ordered decays (f, g) and (g, f) is lost. Appropriate observables are constructed allowing independent experimental determinations of Re( ω\omega ω ) and Im( ω\omega ω ), disentangled from CPT violation in the evolution Hamiltonian Re( θ\theta θ ) and Im( θ\theta θ ). 2σ2\sigma 2σ tensions for both Re( θ\theta θ ) and Im( ω\omega ω ) are shown to be uncorrelated.
The Einstein-Podolsky-Rosen (EPR) entanglement of a neutral K-meson (kaon) pair enjoys the peculiar quantum behaviour of mixing, charge-parity CP{\cal CP}CP violation (C{\cal C}C - charge conjugation, P{\cal P}P - parity) and two non-orthogonal eigenstates of definite time evolution KLK_LKL​ and KSK_SKS​ with very different lifetimes. The dynamics of this strange entanglement, experimentally accessible at the ϕ\phiϕ-factory, makes possible the search for novel phenomena not accessible in any other system, i.e. at interference times and at decoherence times of the single partners after their disentanglement, in order to unveil the nature of the correlation between the two neutral kaons. Until now, the studies have been concentrated on the single kaon intensity distribution between the two decay times t1t_1t1​ and t2t_2t2​ with Δt=t2−t1>0\Delta t = t_2 - t_1 > 0Δt=t2​−t1​>0. Here we show that the entire two single time distributions, before each decay, are physical in both senses: past to future, leading to the state of the living partner at time t2t_2t2​ from the observation of the first decay channel at t1t_1t1​, and future to past, leading to the past state of the decayed kaon at time t1t_1t1​ from the observation of the second decay channel at t2t_2t2​. Our results thus demonstrate an affirmative answer to the title: it does, the past decayed state depends on the result of the future measurement for the living partner. This novel effect is truly observable through the first decay time distributions. Besides the implications for quantum physics for all Δt\Delta tΔt, at large decoherence times the resulting first decayed state is always KSK_SKS​, providing a genuine tag of this kaon state in the decoherence region, and a unique and important experimental tool to prepare a pure KSK_SKS​ beam.
A generalization of the 1935 Einstein-Podolsky-Rosen (EPR) argument for measurements with continuous variable outcomes is presented to establish criteria for the demonstration of the EPR paradox, for situations where the correlation between spatially separated subsystems is not perfect. Two types of criteria for EPR correlations are determined. The first type are based on measurements of the variances of conditional probability distributions and are necessary to reflect directly the situation of the original EPR paradox. The second weaker set of EPR criteria are based on the proven failure of (Bell-type) local realistic theories which could be consistent with a local quantum description for each subsystem. The relationship with criteria sufficient to prove entanglement is established, to show that any demonstration of EPR correlations will also signify entanglement. It is also shown how a demonstration of entanglement between two spatially separated subsystems, if not able to interpreted as a violation of a Bell-type inequality, may be interpreted as a demonstration of the EPR correlations. In particular it is explained how the experimental observation of two-mode squeezing using spatially separated detectors will signify not only entanglement but EPR correlations defined in a general sense.